You only need two lines (four points) to find another point. Two lines will either have 0, 1, or infinitely many intersections. For obvious reasons we will only consider the case where there is one intersection (as the other two cases don't help us pinpoint a specific location).
So we have two lines, let's call them 'line 1' and 'line 2'. These two lines intersect and give us a point, let's call it 'point β'. [Note: I didn't use alpha as it can sometimes look like the letter 'a', which would cause confusion further down]
Let's say you have a third line, we'll call it 'line 3'. Then there are a few possibilities in terms of intersections for line 3:
- a) It lies on top of either line 1 or line 2
- b) It does not intersect any of the original lines
- c) It intersects only one of the original lines (but not in the special way that occurs in case 'a')
- d) It intersects both of the original lines but does not go through point β
- e) It intersects both of the original lines and does go through point β
We can ignore 'a)' since it would mean line 3 is, for all intents and purposes, a copy of one of the first two lines. Which means we essentially only have two lines, not three.
We can ignore 'b)' since it would means line 3 does not intersect the location we are trying to find since it would have zero intersections.
'c)' can also be discarded, but why?
The original point we found, point β, was the intersection of lines 1 and 2. Let's say line 3 also went through point β, then; line 3 would have to intersect both of the original lines. But 'c)' is the possibility where line 3 intersects only one of the original lines; not both. So in possibility 'c)' line 3 intersects a single other line to give us a point, and we know this point can't be point β. Let's call this 'point γ'. Here's the problem; we only want a single location, but the three lines have given us two points; β and γ. Too many locations, hence 'c)' is ignored.
'd)' is also out. Why? Well lines 1 & 2 intersect at point β, and we said possibility 'd)' is where line 3 does not go through point β. This means we now have three points!
- The point where lines 1 & 2 intersect
- The point where lines 2 & 3 intersect
- The point where lines 3 & 1 intersect
We only want one point, but now we have three! This is even worse than possibility 'c)'! So, 'd)' is definitely out.
We have discounted possibilities 'a)' through 'd)', so now we are left with 'e)'. But here's the problem; 'e)' technically has nothing wrong with it, but it's useless! Line 3 intersects the original two lines at a single point, the only place this can happen is at point β. If line 3 does not intersect lines 1 & 2 at point β we are back to possibility 'd)', which gives us three points as we discovered. So now we know, the location is at point β!!! We know this because all three lines intersect at one single point.
But... we already knew where point β was from the first two lines, we even gave that point a name: 'point β'. For line 3 to make sense it has to intersect a point we already found. If it does not then we go back to one of the first four possibilities, which do not make any sense. (except for 'a', which does make sense but is identical to using two lines)
Another way to think about it;
- We have two lines in three dimensional space and we know they both intersect a location that we are trying to find.
- We know that the two lines can be thought of as existing entirely on a two dimensional plane which is a 'slice' of the three dimensional space they exist in. There is only one possible plane where both of these lines exist in their entirety.
The location we want to find must be on that plane. Why? Because;
If the lines exist entirely on the plane then they exist nowhere outside of it, if the location we want to find exists outside of the plane then it exists where no part of the lines exist. If no part of the lines exist where the location is then the lines could not possibly pass through it, but... we know the lines do pass through the location, so that location must lie on the plane.
We all agree that two lines on a plane will intersect to give us a single point. We have two lines which are part of one and only one plane through the 3-d space, they both go through the location, and that location must be on the plane. Therefore, we know exactly where the location is with only two lines even through those two lines are in three dimensional space.
Two lines are all you need.
You may be tempted to think that additional information will give you more 'combinations'. There are, but the problem is; many of them are not possible and the ones that are possible happen to be useless. All the additional combinations belong to one of the five possibilities listed before. If it belongs to;
- 'a)' - One line is a copy of another, so there are only two lines: No additional combinations!
- 'b)' - One of the lines simultaneously does go through the location & does not go through the location. Reductio ad absurdum: No additional possible combinations.
- 'c)' - Gives us too many points, one location somehow exists in more than one location. Reductio ad absurdum: No additional possible combinations.
- 'd)' - Gives us too many points, one location somehow exists in more than one location. Reductio ad absurdum: No additional possible combinations.
- 'e)' - There are a number of different third lines which intersect point β, so we have many new combinations. However, all these additional combinations point to the same location; point β!!!
We definitely do get more combinations from possibility 'e', but none of them give new results! They only give copies of results obtained from two lines intersecting! Therefore; There are more combinations but the same number of locations!
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